Final Review

What to do:

Read lectures 1-12, chapters 1-9 and 12-25

from the textbook

Go over the midterm review

Do the assigned exercises from the textbook

Go over the quiz questions

Use sample exams to practice

Use extra TAs' office hours

Topics to review:

Random variables:

Discrete

o Probability distribution

o Mean and variance of a discrete random variable

Example: Let and be independent and have the same distribution

given below:

-1 0

2/3 1/3

Let . Find the probability distribution of X, E(X) and Var(X).

o Binomial distribution

X ~ Bin (n, p), X = 0, 1, …, n

√

( ) , where (

)

o Geometric distribution

X ~ Geom (p), X = 1, 2, 3, …

√

, where q = 1 – p

Continuous

o Density curve

o Normal distribution

o Normal approximation (sampling distribution for counts and sample

proportions)

Let X be the count of successes in the sample and be the sample

proportion of successes.

When n is large, the sampling distributions of these statistics are

approximately Normal:

X is approximately √

is approximately √

)

As a rule of thumb, we will use this approximation for values of n and p that

satisfy and .

Sampling distribution of sample mean (CLT):

Draw an SRS of size n from any population with mean and finite standard

deviation . When n is large enough,

√

Example: 20% of customers at a bakery will buy a brownie.

(a) What is the probability that more than 110 customers buy a brownie?

(b) How many brownies does the bakery need to have in stock so that the

probability of selling out in a day is 1%?

500 customers arrive at the bakery in a

day. Assume that individual customers

make their purchases independently.

(a) What is the probability that more than 110 customers buy a brownie?

(b) How many brownies does the bakery need to have in stock so that the

probability of selling out in a day is 1%?

Statistical inference

o Confidence intervals

For population mean :

√ if is known

√ if is unknown

For difference of means : √

Pooled: √

where

For single proportion p: √

For difference of proportions :

√

o Hypothesis testing

Test statistic

To test we find statistic

√ ( is known)

To test we find statistic

√ ( is unknown)

To test we find statistic

√

or

√

if

where

To test we find statistic

√

To test we find statistic

where

√ (

) and

P-value

Significance level

Power

Type I and Type II errors

Matched pairs t-procedures

Non-normal populations: sign test

Example: 50 smokers were questioned about the number of hours they sleep

each day. We want to test the hypothesis that the smokers need less sleep than

the general public which needs an average of 7.7 hours of sleep. The population

standard deviation is 0.5

(a) For what values of we would reject the null hypothesis at significance

level of .05.

(b) If the sample mean is 7.5, what can you conclude?

Example: 1500 randomly selected pine trees were tested for traces of the Bark

Beetle infestation. It was found that 153 of the trees showed such traces. Test

the hypothesis that more than 10% of the trees have been infested. (Use a 5%

level of significance)

Example: Suppose that ten identical twins were reared apart.

The mean difference between the high school GPA of the twin brought up in

wealth and the twin brought up in poverty was 0.07. If the standard deviation of

the differences was 0.5, find a 95% confidence interval for the

difference. Assume the distribution of GPA's is approximately normal.

Example: Do employees perform better at work with music playing. The music

was turned on during the working hours of a business with 45 employees. Their

productivity level averaged 5.2 with a standard deviation of 2.4. On a different

day the music was turned off and there were 40 workers. The workers'

productivity level averaged 4.8 with a standard deviation of 1.2. What can we

conclude at the .05 level?

Example: Is the severity of the drug problem in high school the same for boys

and girls? 85 boys and 70 girls were questioned and 34 of the boys and 14 of

the girls admitted to having tried some sort of drug. What can be concluded at

the .05 level?

Example: Consider the following hypothesis test

vs

Assume , , and . Find the probability of a Type II

error for a particular value .